Both sides had early chances to score in the League One clash at Meadow Lane on Wednesday night, but Alan Judge's shot crashed off the bar and Scott Laird's tame effort was smothered.
Darius Charles then stayed on his feet to battle past four defenders before putting a tempting ball across the area that no one in blue was able to convert.
Jeff Hughes was twice denied by Chris Day before Damion Stewart's air-kick gave Chris Beardsley a free run on goal, but the striker snatched his shot wide.
Lee Hughes had a shot saved and substitute Cristian Montano sent a spectacular volley wide, but Stevenage were gifted another chance when the Magpies' defence parted and Don Cowan ran through but fired wide.
But in the 90th minute, County finally got the goal their second-half display merited when Jeff Hughes' shot went through a crowd of players to deceive Day and find the bottom corner.
